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Theorem eldprdi 19930
Description: The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
eldprdi.0 0 = (0g𝐺)
eldprdi.w 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
eldprdi.1 (𝜑𝐺dom DProd 𝑆)
eldprdi.2 (𝜑 → dom 𝑆 = 𝐼)
eldprdi.3 (𝜑𝐹𝑊)
Assertion
Ref Expression
eldprdi (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))
Distinct variable groups:   ,𝐹   ,𝑖,𝐺   ,𝐼,𝑖   0 ,   𝑆,,𝑖
Allowed substitution hints:   𝜑(,𝑖)   𝐹(𝑖)   𝑊(,𝑖)   0 (𝑖)

Proof of Theorem eldprdi
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eldprdi.1 . 2 (𝜑𝐺dom DProd 𝑆)
2 eldprdi.3 . . 3 (𝜑𝐹𝑊)
3 eqid 2724 . . 3 (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)
4 oveq2 7409 . . . 4 (𝑓 = 𝐹 → (𝐺 Σg 𝑓) = (𝐺 Σg 𝐹))
54rspceeqv 3625 . . 3 ((𝐹𝑊 ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)) → ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
62, 3, 5sylancl 585 . 2 (𝜑 → ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
7 eldprdi.2 . . 3 (𝜑 → dom 𝑆 = 𝐼)
8 eldprdi.0 . . . 4 0 = (0g𝐺)
9 eldprdi.w . . . 4 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
108, 9eldprd 19916 . . 3 (dom 𝑆 = 𝐼 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
117, 10syl 17 . 2 (𝜑 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
121, 6, 11mpbir2and 710 1 (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3062  {crab 3424   class class class wbr 5138  dom cdm 5666  cfv 6533  (class class class)co 7401  Xcixp 8887   finSupp cfsupp 9357  0gc0g 17384   Σg cgsu 17385   DProd cdprd 19905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-ixp 8888  df-dprd 19907
This theorem is referenced by:  dprdfsub  19933  dprdf11  19935  dprdsubg  19936  dprdub  19937  dpjidcl  19970
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